Is total mechanical energy conserved? Law of conservation of energy in mechanics
The sum of the kinetic and potential energy bodies that make up a closed system and interact with each other through the forces of gravity and elastic forces, remains unchanged.
This statement expresses law of conservation of energy in mechanical processes . It is a consequence of Newton's laws. Amount E = E k + Ep called full mechanical energy . The law of conservation of mechanical energy is fulfilled only when bodies in a closed system interact with each other by conservative forces, that is, forces for which the concept of potential energy can be introduced.
Ticket 11
Expression of the angular momentum of a body with one fixed point in terms of the matrix of moments of inertia of the body.
It has solid, one of the points of which is fixed. The motion of the body is considered relative to some coordinate system О xyz .
Kinetic moment relative to a fixed point:
Where r k is the radius vector of some point of the body. m k - point mass. V k - the speed of this point relative to the selected reference system.
euler formula
In projections on the axis:
For the projection of the angular momentum on the O x axis, taking into account (2'), we have: 
The sums in (1') are respectively the axial and centrifugal moments of inertia. We get: 
According to (3), the projections on the coordinate axes of the angular momentum of the body relative to its fixed point are calculated. The angular momentum according to the projections is determined by the formula: 
For fixed axes, the axial and centrifugal moments of inertia change with the rotation of the body and, therefore, depend on time due to a change in the position of the body relative to these axes.
If we apply the inertia tensor: 
And take into account the rule of multiplication of the tensor by the vector column omega, then it can be briefly expressed by the formula: .
We simplify formula (3) for projections:
In this case, the projections of the angular momentum are calculated in the same way as in the case if each of the main axes of inertia were a fixed axis of rotation of the body. The principal axes of inertia for the fixed point O are usually movable axes attached to the rotating body itself. Only such axes can be principal during the entire time of rotation of the body. Other moving or fixed axes can only be master at certain times.
Kinetic energy of translational motion
The kinetic energy of the system is called the scalar value T, equal to the arithmetic sum of the kinetic energies of all points of the system
Kinetic energy is a characteristic of both the translational and rotational motion of the system, therefore, the theorem on the change in kinetic energy is especially often used in solving problems.
If the system consists of several bodies, then its kinetic energy is obviously equal to the sum of the kinetic energies of these bodies:
Kinetic energy is a scalar and always positive quantity.
Let's find formulas for calculating the kinetic energy of the body in different occasions movement.
1. translational movement. In this case, all points of the body move at the same speed, equal to the speed of the center of mass. That is, for any point
Thus, the kinetic energy of a body in translational motion is equal to half the product of the body's mass times the square of the center of mass velocity. Off direction value T does not depend.
Ticket 12
Differential equation of rotation of a rigid body around a fixed axis
The differential equation has the form:
, (2.6)
Where 
is the angular acceleration of the body.
Equation (2.6) is obtained from equation (2.4) of the theorem by substituting formula (2.3) into it.
(2.3)
(2.4)
By integrating equation (2.6), one can determine the law of body rotation. Methods for solving such problems:
- depict the body in an arbitrary position; show the external forces acting on the body; we show the axis directed along the axis of rotation of the body in the direction from which the rotation is seen to occur counterclockwise;
- we find the sum of the moments of external forces about the axis;
- we calculate, if not set, the moment of inertia of the body;
- compose equation (2.6), integrating this equation, we determine the law of rotation of the body.
POTENTIAL FORCES
A field of forces that remains constant in time is called stationary. In a stationary force field, the force acting on a particle depends only on its position. The work done by the field forces when moving a particle from point 1 to point 2 depends, generally speaking, on the trajectory along which the particle moves from the initial position to the final one. At the same time, there are stationary force fields in which the work done on the particles by the field forces does not depend on the shape of the trajectory between points 1 and 2. Forces with this property are called potential or conservative, and the corresponding force field is called a potential field. An example of potential forces are elastic forces, gravity.
ticket 13 1. Plane-parallel (or flat) is such a motion of a rigid body, at which all its points move parallel to some fixed plane P.  Consider a section of the body by some plane OXY parallel to the fixed plane P (Fig. 1). to the plane P, move identically. Therefore, to study the motion of the whole body, it is sufficient to study how the section of the body moves in the OXY plane. In the future, we will combine the OXY plane with the plane of the figure, and instead of the whole body, depict only its section. The position of the section in the OXY plane is determined by the position of some segment AB drawn in this section (Fig. 2). The position of the segment AB can be determined by knowing the coordinates of the point A and the angle that the segment AB forms with the x-axis. Point A, chosen to determine the position of the section, is called the pole. When the body moves, the quantities and will change: (1.74) The equations that determine the law of the ongoing movement are called equations of plane-parallel motion solid body. 2. The main moment of all internal forces of the system (relative to any chosen center) at any time is equal to zero (M O i =0). M-vector. or . The internal forces will be balanced when the system under consideration is an absolutely rigid body. Indeed, if we take an arbitrary center ABOUT, then from Fig. it's clear that  .
 | ticket 14 1. The kinetic energy of the system is the sum of the kinetic energies of all material points included in the system; at forward movement: E=mV 2 /2; when rotating around a fixed axis: E=I Z v 2 /2; in plane-parallel motion: E=mV C 2 /2-I Z v 2 /2, where V C is the speed of the center of mass, v is the angular velocity. Kinetic energy mechanical system is the energy of the center of mass movement plus the energy of movement relative to the center of mass: E=E 0 +E R , where E is the total kinetic energy of the system, E 0 is the kinetic energy of the center of mass movement, E R is the relative kinetic energy of the system. In other words, the total kinetic energy of a body or system of bodies in complex motion is equal to the sum of the system's energy in translational motion and the system's energy in its spherical motion relative to the center of mass. 2. Degrees of freedom is a set of independent coordinates of movement and / or rotation, which completely determines the position of the system or body (and together with their time derivatives - the corresponding speeds - completely determines state mechanical system or body - that is, their position and movement). Generalized coordinates (d.c.) systems are called such quantities that generalize several independent Cartesian coordinates into angles, linear distances, areas. The convenience lies in the fact that o.k. can be chosen taking into account superimposed relationships, i.e. in accordance with the nature of the movement allowed for the system by the totality of the imposed connections. |
Ticket
1) For the internal forces of a mechanical system, the property takes place: the main vector and main point internal forces of the mechanical system are equal to zero.
.
This follows from the fact that internal forces are forces of interaction between the points of the system, which are pairwise equal and directed in opposite directions.
2) If all forces of the system are potential, then the generalized forces of the system are expressed in terms of the potential energy of the system as Q j \u003d -dP / q j, and the Lagrange equations of the second kind can be written in the form
Since the potential energy does not depend on generalized velocities, then. Let us introduce the function
Ticket 16.
1. Theorem on the change in the kinetic energy of a mechanical system in differential form
The change in the kinetic energy of a mechanical system at some of its displacement is equal to the sum of the work of external and internal forces applied to the points of the system at the same displacement.
2. Holding and stationary links
If the function depends explicitly on time, then the relationship is said to be non-stationary or rheonomic; if this function does not explicitly depend on time, then they say that this connection is stationary or scleronoma.
If the connection is given by equality, then they say that such a connection is holding back or bilateral:
Ticket 17
1 Theorem on the change in the kinetic energy of a mechanical system
The kinetic energy of a system is the sum of the kinetic energies of all the bodies in the system. For the quantity defined in this way, the statement is true:
The change in the kinetic energy of the system is equal to the work of all internal and external forces acting on the bodies of the system.
2 Holonomic constraints
Holonomic connection- mechanical connection, which imposes restrictions only on the positions (or displacements) of points and bodies of the system.
Mathematically expressed as an equality:
Ticket 18
1.Principle of Euler-D'Alembert for a material point
According to this principle, for each i-th point of the system, the equality is true, where is the active force acting on this point, is the reaction of the connection imposed on the point, is the force of inertia, numerically equal to the product of the mass of the point and its acceleration and directed opposite to this acceleration ()
2 kinetic energy of the body in plane motion
Ticket 19
Equations of kinetostatics.
Kinetostatics- a branch of mechanics that deals with methods for solving dynamic problems using analytical or graphical methods of statics. K. is based on the d "Alembert principle, according to which the equations of motion of bodies can be compiled in the form of equations of statics, if inertial forces are added to the forces actually acting on the body and the reactions of the bonds. K. methods are used in solving a number of dynamic problems, especially in dynamics machines and mechanisms.
kinetostatics equations for a material point:
where F, R, Ф are the main vectors of active forces, reactions of bonds and forces of inertia;
Fz, Rz, Ф z - the main moments of active forces, reactions of bonds and inertia forces relative to the point O 1
With the existing closed mechanical system, the bodies interact through the forces of gravity and elasticity, then their work is equal to the change in the potential energy of the bodies with the opposite sign:
A \u003d - (E p 2 - E p 1) .
Following from the kinetic energy theorem, the work formula takes the form
A \u003d Ek 2 - Ek 1.
Hence it follows that
E k 2 - E k 1 \u003d - (E p 2 - E p 1) or E k 1 + E p 1 \u003d E k 2 + E p 2.
Definition 1The sum of the kinetic and potential energy of bodies, constituting a closed system and interacting with each other through the forces of gravity and elastic forces, remains unchanged.
This statement expresses the law of conservation of energy in a closed system and in mechanical processes, which is a consequence of Newton's laws.
Definition 2
The law of conservation of energy is fulfilled when forces interact with potential energies in a closed system.
Example N
An example of the application of such a law is the determination of the minimum strength of the light inextensibility of a thread that holds an adze with a mass m, rotating it vertically relative to the plane (the Huygens problem). The detailed solution is shown in Figure 1. 20 . 1 .
Picture 1 . 20 . 1 . To the Huygens problem, where F → is taken as the force of the thread tension at the lower point of the trajectory.
The record of the total energy conservation law at the upper and lower points takes the form
m v 1 2 2 = m v 2 2 2 + m g 2 l .
F → is located perpendicular to the speed of the body, hence the conclusion that it does not do work.
If the rotation speed is minimal, then the tension of the thread at the top point is zero, which means that centripetal acceleration can only be reported using gravity. Then
m v 2 2 l = m g .
Based on the relations, we get
v 1 m i n 2 = 5 g l .
Centripetal acceleration is created by forces F → and m g → with opposite directions relative to each other. Then the formula will be written:
m v 1 2 2 = F - m g .
It can be concluded that at the minimum body speed at the top point, the thread tension will be equal in absolute value to the value F = 6 m g .
Obviously, the strength of the thread must exceed the value.
Using the law of conservation of energy through the formula, you can get the relationship between the coordinates and velocities of the body at two different points of the trajectory, without using the analysis of the law of motion of the body at all intermediate points. This law makes it possible to significantly simplify the solution of problems.
Real conditions for moving bodies involve the action of gravity, elasticity, friction and resistance of the given medium. The work of the friction force depends on the length of the path, so it is not conservative.
Definition 3Friction forces act between the bodies that make up a closed system, then mechanical energy is not preserved, its part passes into the internal. Any physical interactions do not provoke the emergence or disappearance of energy. It changes from one form to another. This fact expresses the fundamental law of nature - law of conservation and transformation of energy.
The consequence is the statement about the impossibility of creating a perpetual motion machine (perpetuum mobile) - a machine that would do work and not consume energy.
Picture 1 . 20 . 2. Perpetual motion project. Why won't this machine work?
Exists a large number of such projects. They do not have the right to exist, since some design errors of the entire device are clearly visible in the calculations, while others are masked. Attempts to implement such a machine are futile, since they contradict the law of conservation and transformation of energy, so finding a formula will not give results.
If you notice a mistake in the text, please highlight it and press Ctrl+Enter
Theory: Energy does not disappear anywhere, it turns from one type into another, and it does not arise from nowhere.
Energy can be converted into mechanical work or into.
The total energy of a closed system is a constant value: E \u003d E to + E p
For example: a body of mass 2 kg will be raised to a height of 1 meter, at this height potential body E p \u003d mgh \u003d 20 J, as the body falls, the height decreases, the potential energy also decreases. At the same time, the speed of the body begins to increase, as a result of which the kinetic energy also increases. It turns out that the energy goes from potential to kinetic. At the moment of touching the surface, the potential energy is zero, the kinetic energy is maximum and equal to 20 J in the same way as at the beginning. If the body is elastically reflected, then as it rises to a height, the kinetic energy will decrease and turn into potential energy. 
Tasks: A ball is thrown vertically upward from the surface of the earth. Air resistance is negligible. With an increase in the initial speed of the ball by 2 times, the height of the ball
1) will increase by √ 2 times
2) will increase by 2 times
3) will increase by 4 times
4) will not change
Exercise: A bullet moving at a speed of 600 m/s pierced a board 1.5 cm thick and had a speed of 300 m/s at the exit from the board. Determine the mass of the bullet if the average drag force acting on the bullet in the plank is 81 kN.
A body of mass m is thrown vertically upwards from the earth. initial speedυ 0 , rose to the height h 0 . Air resistance is negligible. The total mechanical energy of the body at some intermediate height h is 
Solution: Since air resistance is negligible, therefore total energy system does not change. The total mechanical energy of the body at some intermediate height h is equal to the energy at the maximum height mgh 0 .
Answer: 2
OGE task in physics (fipi): The ball moves down the inclined chute without friction. Which of the following statements about the ball's energy is true for such a motion?
1) The kinetic energy of the ball increases, its total mechanical energy does not change.
2) The potential energy of the ball increases, its total mechanical energy does not change.
3) Both the kinetic energy and the total mechanical energy of the ball increase.
4) Both the potential energy and the total mechanical energy of the ball decrease.
Solution: As you move down, the speed of the ball increases. Therefore, the kinetic energy increases. Since there is no friction, and the system can be considered closed, the total mechanical energy does not change.
Answer: 1
OGE task in physics (fipi): A freight car moving along a horizontal track at low speed collides with another car and stops. This compresses the buffer spring. Which of the following energy transformations occurs in this process?
1) the kinetic energy of the car is converted into the potential energy of the spring
2) the kinetic energy of the car is converted into its potential energy
3) the potential energy of the spring is converted into its kinetic energy
4) the internal energy of the spring is converted into the kinetic energy of the car
Solution: At first the car was moving, so it had kinetic energy. During the collision, the spring was compressed, i.e. the kinetic energy of the car is converted into the potential energy of the spring
1.7. LAW OF CONSERVATION OF MECHANICAL ENERGY
Formulation of the law of conservation of mechanical energy. Formulation in the case of the presence of dissipative forces. Graphical representation of energy. Finite and infinite motions. Absolutely resilient hit. Absolutely inelastic impact.
Total mechanical energy of the system- energy of mechanical movement and interaction, i.e. is equal to the sum of kinetic and potential energies. The law of conservation of mechanical energy: in a system of bodies between which only conservative forces the total mechanical energy is conserved, i.e. does not change over time. This - fundamental law of nature. He is a consequence uniformity of time - invariance of physical laws with respect to the choice of the origin of time. All forces in mechanics are usually divided into conservative And non-conservative. Conservative forces are called, the work of which does not depend on the shape of the trajectory (path) between two points, but depends only on the initial and final positions of the body relative to another. In other words, the work of conservative forces along a closed trajectory is zero. An example of conservative forces are gravity, elastic force, etc. First of all, they are dissipative forces(converting mechanical energy into other forms of energy), for example, the force of friction. If there change, then it is equal to the work of dissipative forces. finite– movement of points in a limited area of space. Infinite- the body goes to infinity. Absolutely elastic impact - a collision of two bodies, as a result of which no deformations remain in both interacting bodies and all the kinetic energy that the bodies possessed before the impact is again converted into kinetic energy after the impact. conservation laws momentum and conservation of mechanical energy performed . Absolutely inelastic impact - the collision of two bodies, as a result of which the bodies are combined, moving on as a single body. Not fulfilled the law of conservation of mechanical energy: due to deformation, part of the kinetic energy passes into the internal energy of bodies (heating).
Let us introduce the concept of the total mechanical energy of a particle. The increment of the kinetic energy of the particle is equal to the elementary work of the resultant of all forces acting on the particle. If a particle is in a potential field, then a conservative force acts on it from this potential field. In addition, other forces of a different origin can also act on the particle. Let's call them outside forces .
Thus, the resultant of all forces acting on a particle can be represented as . The work of all these forces is used to increase the kinetic energy of the particle:
According to (6.7), the work of the field forces is equal to the decrease in the potential energy of the particle, i.e. . Substituting this expression into the previous one and moving the term to the left, we get
From this it can be seen that the work of external forces goes to the increment of the value . This quantity - the sum of kinetic and potential energy - is called total mechanical energy of the particle in the field :
on the final move from point 1 to point 2
|
|
(7 .3) |
those. the increment of the total mechanical energy of a particle on a certain path is equal to the algebraic sum of the work of all external forces, acting on a particle along the same path. If , then the total mechanical energy of the particle increases; if , then it decreases.
The total mechanical energy of a particle can change only under the action of external forces. This directly implies the law of conservation of the total mechanical energy of a particle in external field: if external forces are absent or such that the algebraic sum of their powers is equal to zero during the time of interest to us, then the total mechanical energy of the particle remains constant during this time. In other words,
|
(7 .4) |
Already in such a simple form this law saving makes it quite easy to get answers to a number of important issues without involving the equations of motion, which, as we know, is often associated with cumbersome and tedious calculations. It is this circumstance that makes conservation laws a very effective research tool.
Let us illustrate the possibilities and advantages that the application of the conservation law (7.4) gives by the following example.
Example. Let a particle move in a one-dimensional potential fieldU(x. If there are no external forces, then the total mechanical energy of a particle in a given field, i.e., E, does not change in the process of motion, and we can simply solve, for example, such questions as:
1. Determine, without solving the basic equation of dynamics,v(x) - speed of a particle depending on its coordinates. To do this, it is enough to know, according to the equation(7.4) , a specific form of the potential curveU(x) and the value of the total energy E (the right side of this equation).
2. Determine the region of change of the x-coordinate of the particle, in which it can be located at a given value of the total energy E. It is clear that in the region whereU> E, the particle cannot get in, because the potential energyUparticle cannot exceed its total energy. It immediately follows from this that when (Fig. 7.1) the particle can move in the region
between coordinates (oscillates) or to the right of the coordinate . However, a particle cannot go from the first region to the second (or vice versa): this is prevented by a potential barrier separating both these regions. Note that when a particle moves in a limited region of the field, it is said that it is in a potential well, in our case - between .
The particle behaves differently when (Fig. 7.1): the entire area to the right is available to it . If in initial moment the particle was at the point , then it will continue to move to the right. Determining the change in the kinetic energy of a particle depending on its position x can serve as a useful independent exercise.
So far, we have limited ourselves to the behavior one particles with energy point vision. Now let's move on to particle system. It can be any body, gas, any mechanism, solar system etc.
In the general case, the particles of the system can interact both with each other and with bodies that are not included in the given system. A system of particles, which is not affected by any foreign bodies or their influence is negligible, is called closed or isolated. The concept of a closed system is a natural generalization of the concept of an isolated material point and plays important role in physics.
Let us introduce the concept of potential energy of a system of particles. Let us consider a closed system, between the particles of which only central forces act, i.e., forces that depend on this character interactions only on the distance between them and directed along the straight line connecting them.
Let us show that in any reference frame, the work of all these forces during the transition of a system of particles from one position to another can be represented as a decrease in a certain function that, for a given character of interaction, depends only on the configuration of the system itself or on the relative arrangement of its particles. We will call this function own potential energy of the system, as opposed to external potential energy that characterizes the interaction of a given system with other bodies.
Let us first consider a system of two particles. Let us calculate the elementary work of the forces with which these particles interact with each other. Let in an arbitrary reference frame at some point in time the position of particles be determined by the radius vectors and . If during the time dt the particles moved and respectively, then the work of the interaction forces and is equal to
Now we take into account that, according to Newton's third law, therefore, the previous expression can be rewritten as follows:
We introduce a vector characterizing the position of the 1st particle relative to the 2nd. Then 
and after substitution into the expression for work, we get
.
The force is central, therefore the work of this force is equal to the decrease in the potential energy of interaction of a given pair of particles, i.e.
Since the function depends only on the distance between the particles, it is clear that the work does not depend on the choice of reference frame.
Let us now consider a system of three particles, since the result obtained in this case can easily be generalized to a system of an arbitrary number of particles. The elementary work performed by all interaction forces during the elementary displacement of all particles can be represented as the sum of the elementary works of all three pairs of interactions, i.e.
But for each pair of interactions, as shown 
, That's why
where is the function own potential energy given system of particles:
Since each term in this sum depends on the distance between the corresponding particles, it is obvious that the self-potential energy U of a given system depends on the relative arrangement of particles at the same time, or, in other words, on the configuration of the system.
Similar reasoning is also valid for a system of any number of particles. Therefore, it can be argued that each configuration of an arbitrary system of particles has its own potential energy U , and the work of all central internal forces with a change in the configuration of the system is equal to the decrease in the system's own potential energy, i.e.
|
(7 .5) |
and with a finite displacement of all particles of the system
|
(7 .6) |
where and are the values of the potential energy of the system in the initial and final states.
Self potential energy of the system U is a non-additive quantity, i.e. it is not equal in general case the sum of its own potential energies of its parts. It is also necessary to take into account the potential energy of interaction of individual parts of the system
|
|
(7 .7) |
,
where is the self potential energy of a part of the system.
It should also be borne in mind that the system's own potential energy, as well as the potential energy of the interaction of each pair of particles, is determined up to the addition of an arbitrary constant, which, however, is completely insignificant here as well.
In conclusion, we present useful formulas for calculating the self-potential energy of the system. First of all, we show that this energy can be represented as
|
(7 .8) |
where is the potential energy of particle interaction with all other particles of the system. Here the sum is taken over all particles of the system. Let us verify the validity of this formula first for a system of three particles. It was shown above that the self-potential energy of this system 
Let's transform this sum as follows. We represent each term in a symmetrical form: 
, because it is clear that . Then
Let's group the members with the same first index:
Each sum in parentheses represents the potential energy of the particle's interaction with the other two. So the last expression can be rewritten like this:
which completely corresponds to formula (7.8).
The generalization of the result obtained to an arbitrary system is obvious, because it is clear that such reasoning is completely independent of the number of particles that make up the system.
For a system whose interaction between particles is of a gravitational or Coulomb nature, formula (7.8) can also be transformed to another form, using the concept of potential. Let us replace in (7.8) the potential energy of a particle by the expression , where is the mass (charge) of the particle, and is the potential created by all other particles of the system at the point where the particle is located.
where is the bulk density of mass or charge, is the volume element. Here, integration is carried out over the entire volume occupied by masses or charges.
Let us classify forces according to their properties. It is known that the particles of the system under consideration can interact both with each other and with bodies that are not included in this system. In accordance with this, the forces of interaction between the particles of the system are called internal , and the forces due to the action of other bodies that are not included in this system - external. In a non-inertial frame of reference, the forces of inertia must also be attributed to the latter.
In addition, all forces are divided into potential And non-potential . Potential forces are forces that, for a given nature of interaction, depend only on the configuration of the mechanical system. The work of these forces, as has been shown, is equal to the loss of the potential energy of the system. The so-called non-potential forces are dissipative forces are the forces of friction and resistance, and also energy forces that cause an increase in the mechanical energy of the system due to other types of energy (for example, the explosion of an artillery shell). An important feature given forces is that the total work internal dissipative forces of the system under consideration is negative, and energy forces - positive, and in any frame of reference. Let us prove this for dissipative forces.
Any dissipative force can be represented as
|
(7 . 1 4) |
where is the speed of a given body relative to another body (or medium) with which it interacts; is a positive coefficient that generally depends on the speed . The force is always directed opposite to the vector. Depending on the choice of reference system, the work of this force can be either positive or negative. The total work of all internal dissipative forces is always negative . Turning to the proof of this, we note first of all that the internal dissipative forces in a given system will occur in pairs, and in each pair, according to Newton's third law, they are identical in absolute value and opposite in direction. Let us find the elementary work of an arbitrary pair of dissipative forces of interaction between bodies 1 And 2 in the reference frame, where the velocities of these bodies in this moment are equal :
Now we take into account that 
- body speed 1
relative to the body 2
, and also that 
. Then the expression for work is transformed as follows:
This shows that the work of an arbitrary pair of internal dissipative forces of interaction is always negative, and hence the total work of all pairs of internal dissipative forces is also always negative. Thus, indeed,
|
(7 . 1 5) |
Now we can formulate the law of conservation of the total mechanical energy of a system of particles. It was shown above that the increment in the kinetic energy of the system is equal to the work done by All forces acting on All system particles. Dividing these forces into external and internal, and internal, in turn, into potential and non-potential, we write the previous statement as follows:
Now we take into account that the work of internal potential forces is equal to the decrease in the system's own potential energy, i.e. 
Then the previous expression will take the form
Obviously the energy E depends on the velocities of the particles of the system, the nature of the interaction between them, and the configuration of the system. In addition, the energy E, like potential energy U, is determined up to the addition of an insignificant arbitrary constant and is the quantity non-additive , i.e. energy E system is not equal in the general case to the sum of its energies separate parts. According to (7.7)
|
|
(7 . 1 8) |
where is the mechanical energy of a part of the system, is the potential energy of the interaction of its individual parts.
Let us return to formula (7.16). Let us rewrite it, taking into account (7.17), in the form
In all phenomena occurring in nature, energy does not arise and does not disappear. It only changes from one species to another, while its value is preserved.
Law of energy conservation- the fundamental law of nature, which consists in the fact that for an isolated physical system a scalar physical quantity can be introduced, which is a function of the parameters of the system and is called energy, which is conserved over time. Since the law of conservation of energy does not refer to specific quantities and phenomena, but reflects a general pattern that is applicable everywhere and always, it can be called not a law, but the principle of conservation of energy.
Law of conservation of mechanical energy
In mechanics, the law of conservation of energy states that in a closed system of particles, the total energy, which is the sum of kinetic and potential energy and does not depend on time, that is, is the integral of motion. The law of conservation of energy is valid only for closed systems, that is, in the absence of external fields or interactions.
The forces of interaction between bodies for which the law of conservation of mechanical energy is fulfilled are called conservative forces. The law of conservation of mechanical energy is not satisfied for friction forces, since in the presence of friction forces, mechanical energy is converted into thermal energy.
Mathematical formulation
The evolution of a mechanical system of material points with masses \(m_i\) according to Newton's second law satisfies the system of equations
\[ m_i\dot(\mathbf(v)_i) = \mathbf(F)_i \]
Where
\(\mathbf(v)_i \) are the velocities of material points, and \(\mathbf(F)_i \) are the forces acting on these points.
If forces are given as the sum of potential forces \(\mathbf(F)_i^p \) and nonpotential forces \(\mathbf(F)_i^d \) , and the potential forces are written as
\[ \mathbf(F)_i^p = - \nabla_i U(\mathbf(r)_1, \mathbf(r)_2, \ldots \mathbf(r)_N) \]
then, multiplying all the equations by \(\mathbf(v)_i \) we can get
\[ \frac(d)(dt) \sum_i \frac(mv_i^2)(2) = - \sum_i \frac(d\mathbf(r)_i)(dt)\cdot \nabla_i U(\mathbf(r )_1, \mathbf(r)_2, \ldots \mathbf(r)_N) + \sum_i \frac(d\mathbf(r)_i)(dt) \cdot \mathbf(F)_i^d \]
The first sum on the right side of the equation is nothing more than the time derivative of a complex function, and therefore, if we introduce the notation
\[ E = \sum_i \frac(mv_i^2)(2) + U(\mathbf(r)_1, \mathbf(r)_2, \ldots \mathbf(r)_N) \]
and call this value mechanical energy, then, integrating the equations from time t=0 to time t, we can obtain
\[ E(t) - E(0) = \int_L \mathbf(F)_i^d \cdot d\mathbf(r)_i \]
where the integration is carried out along the trajectories of motion of material points.
Thus, the change in the mechanical energy of a system of material points over time is equal to the work of nonpotential forces.
The law of conservation of energy in mechanics is valid only for systems in which all forces are potential.
The law of conservation of energy for an electromagnetic field
In electrodynamics, the law of conservation of energy is historically formulated in the form of Poynting's theorem.
The change in the electromagnetic energy contained in a certain volume over a certain time interval is equal to the flow of electromagnetic energy through the surface that limits this volume, and the amount of thermal energy released in this volume, taken with the opposite sign.
$ \frac(d)(dt)\int_(V)\omega_(em)dV=-\oint_(\partial V)\vec(S)d\vec(\sigma)-\int_V \vec(j)\ cdot \vec(E)dV $
The electromagnetic field has energy that is distributed in the space occupied by the field. When the field characteristics change, the energy distribution also changes. It flows from one region of space to another, perhaps changing into other forms. Law of energy conservation for an electromagnetic field is a consequence of the field equations.
Inside some closed surface S, limiting the amount of space V occupied by the field contains energy W is the energy of the electromagnetic field:
W=Σ(εε 0 E i 2 / 2 +μμ 0 H i 2 / 2)ΔV i .
If there are currents in this volume, then the electric field produces work on moving charges, per unit time equal to
N=Σ ij̅ i ×E̅ i . ΔV i .
This is the amount of field energy that goes into other forms. It follows from Maxwell's equations that
∆W + N∆t = -∆t∮ SS̅ × n̅ . da,
Where ∆W is the change in the energy of the electromagnetic field in the volume under consideration over time Δt, a vector S̅ = E̅ × H̅ called the Poynting vector.
This law of conservation of energy in electrodynamics.
Through a small area ΔA with unit normal vector n̅ per unit of time in the direction of the vector n̅ flowing energy S̅ × n̅ .ΔA, Where S̅- meaning Pointing vectors within the site. The sum of these quantities over all elements of a closed surface (denoted by the integral sign), which is on the right side of the equality , is the energy flowing out of the volume bounded by the surface per unit time (if this value is negative, then energy flows into the volume). Pointing vector determines the energy flow of the electromagnetic field through the area, it is non-zero everywhere where the vector product of the electric and magnetic field strength vectors is non-zero.
There are three main directions practical application electricity: transmission and transformation of information (radio, television, computers), transmission of momentum and momentum (electric motors), transformation and transmission of energy (electric generators and power lines). Both momentum and energy are transferred by the field through empty space, the presence of a medium leads only to losses. Energy is not transmitted through wires! Wires with current are needed to form electric and magnetic fields of such a configuration that the energy flow, determined by the Poynting vectors at all points in space, is directed from the energy source to the consumer. Energy can be transmitted without wires, then it is carried by electromagnetic waves. ( Internal energy The sun is waning, being carried away by electromagnetic waves, mostly by light. Some of this energy sustains life on Earth.)
Javascript is disabled in your browser.ActiveX controls must be enabled in order to make calculations!
